Problem 37 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ (p q)^{8} $$

Step-by-Step Solution

Answer

The simplified form is \( p^8 q^8 \).

Step 1: Identify the Rule to Use

The expression \((pq)^8\) involves a product raised to a power. We can apply the Power of a Product Rule, which means that \((ab)^n = a^n b^n\), to simplify it.

Step 2: Apply the Power of a Product Rule

Using the Power of a Product Rule on \((pq)^8\), we apply the rule separately to each variable in the product: \[(pq)^8 = p^8 q^8\].

Step 3: Write the Simplified Expression

The simplified form of the expression \((pq)^8\) is \(p^8 q^8\). This means we have distributed the exponent 8 to both p and q.

Key Concepts

Power of a Product

The **power of a product** rule helps in simplifying expressions where a product is raised to a power. For any two numbers or variables, say \(a\) and \(b\), raised to an exponent \(n\), the rule can be stated as: \((ab)^n = a^n b^n\). This means that both \(a\) and \(b\) will be raised to the power \(n\) individually.
This rule is derived from the basic properties of exponents, which state that multiplication distributes over exponents. In practical terms, if you have an expression such as \((xy)^5\), you apply the power to each component of the product separately, resulting in \(x^5 y^5\).
Using this rule properly can save a lot of time and effort in simplifying algebraic expressions. Make sure to always apply the power to each part of the product.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form so they are easier to work with. When using the power of a product rule, we simplify by distributing the power to each part of the product. For instance, with the expression \((pq)^8\), we apply the exponent 8 to both \(p\) and \(q\), resulting in \(p^8 q^8\).
Simplification can often lead to more manageable equations or identities and can help further calculations or even comparisons between different expressions.
When simplifying, it is crucial to ensure that the rules of exponents are applied consistently to avoid mistakes. Always verify each step for accuracy as you distribute powers and manipulate expressions.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number being multiplied, and the exponent indicates how many times the base is used as a factor. For example, \(a^n\) means the base \(a\) is multiplied by itself \(n\) times.
Understanding exponentiation is key to mastering more complex algebraic functions and calculus concepts. It forms the foundation for applying the power of a product rule and many other exponent-related rules.
Notably, exponentiation follows specific properties and rules including:

• \((a^m)^n = a^{m \cdot n}\)
• \(a^m\cdot a^n = a^{m + n}\)
• \(\frac{a^m}{a^n} = a^{m-n}\)

These properties are essential for breaking down expressions like \((pq)^8 = p^8 q^8\) and for understanding how various algebraic transformations are reached through exponentiation.